The remaining two direct forms are obtained by formally
transposingdirect-forms I and II
[60, p. 155]. Filter transposition may also be called
flow graph reversal, and transposing a
Single-Input, Single-Output (SISO) filter does not alter its transfer
function. This fact can be derived as a consequence of
Mason's gain formula for signal flow graphs [49,50]
or Tellegen's theorem (which implies that an LTIsignal flow
graph is interreciprocal with its transpose)
[60, pp. 176-177]. Transposition of filters in
state-space form is discussed in §G.5.

The transpose of a SISO digital filter is quite
straightforward to find: Reverse the direction of all signal paths, and
make obviously necessary accommodations. ``Obviously necessary
accommodations'' include changing signal branch-points to summers, and
summers to branch-points. Also, after this operation, the input
signal, normally drawn on the left of the signal flow graph, will be
on the right, and the output on the left. To renormalize the layout,
the whole diagram is usually left-right flipped.

Figure 9.3 shows the Transposed-Direct-Form-I (TDF-I) structure for the
general second-order IIR digital filter, and
Fig.9.4 shows the
Transposed-Direct-Form-II
(TDF-II) structure.
To facilitate comparison of the transposed with the original, the
input and output signals remain ``switched'', so that signals
generally flow right-to-left instead of the usual left-to-right.
(Exercise: Derive forms TDF-I/II by transposing the DF-I/II
structures shown in
Figures 9.1 and 9.2.)

Figure:
Transposed-Direct-Form-I implementation of a
second-order IIR digital filter. Note that the input signal comes in
from the right, and the output is on the left. Compare to Fig.9.1.
The four ``state variable'' signals are labeled arbitrarily as
through
.

Figure:
Transposed-Direct-Form-II implementation of a
second-order IIR digital filter (input on the right, output on the
left). Compare to Fig.9.2.